screening 110320 jpe
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Active Screening in Insurance Markets
Shinichi Kamiya
Nanyang Business SchoolNanyang Technological University
Nanyang Avenue, Singapore 639798
Mark J. Browne
Wisconsin School of Business
University of Wisconsin-Madison975 University Avenue
Madison, Wisconsin 53706-1323
Submitted to Journal of Public EconomicsMarch 21, 2011
Corresponding author, Email: [email protected], Tel.: +65 6790 5718, Fax: +65 6791 3697
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Active Screening in Insurance Markets
Abstract
In a market characterized by asymmetric information, a party with private informa-
tion may reveal the information through its self-selection, the mechanism of which has
received considerable attention in the academic literature. The party might also reveal
the private information through a revelation test offered by an uninformed party. We
consider competitive insurance markets in which insurers induce information revelation
with self-selection mechanisms and risk classification tests. We demonstrate that con-
ditional and unconditional contracts may coexist in equilibrium when the conditional
contract offered by an insurer has an accuracy level that is superior to that of other
insurers. We also show that when multiple firms use tests that are similar in terms of
their accuracy, conditional contracts do not hold in a Nash equilibrium.
Keywords: test, screening, adverse selection, insurance
JEL Codes: D81, D82, G22
1 Introduction
Self-selection models in markets characterized by asymmetric information have received con-
siderable attention in the literature. Sorting of informed parties in these models occurs with
different types choosing different contract from menus offered by competing uninformed par-
ties. In the context of insurance markets, uninformed insurance companies compete against
each other in terms of the premium and coverage that they offer to informed individuals who
then self-select. Self-selection results in a decrease in information asymmetry. In contrast
to the attention paid to self-selecting models, the literature has paid little attention to an-
other widely observed market response to information asymmetry, risk classification through
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testing, with which uninformed parties actively screen informed parties. Underwriting in
financial markets, interviewing in job markets, and dating in the marriage market are all
examples. In the current study, we analyze equilibrium in a market in which asymmetric
information is reduced through self-selection and through testing. In practice, testing tends
to be imperfect, and noisy tests may entail pooling contracts. We consider when contracts
conditional on a test result are offered in insurance transactions and whether such conditional
contracts hold in equilibrium.
In our construct, insurers have a choice to offer either a single unconditional contract that
allows individuals to self-select or a single conditional contract that requires individuals to
take a test first that reveals whether they are high risks or low risks. Individuals have a choice
between self-selecting the unconditional contract and taking a test to purchase a conditional
contract. In the absence of the risk classification associated with a conditional contract, the
market is characterized by a Rothschild-Stiglitz (RS, hereafter) Nash equilibrium (Rothschild
and Stiglitz, 1976).
We demonstrate that individuals are willing to deviate from a self-selection contract to a
conditional contract if the test used for the conditional contract is relatively accurate. Equi-
librium conditions are investigated under several different market assumptions, and we show
that both conditional and unconditional contracts can coexist in several circumstances. In
contrast, it is also shown that conditional contracts are not sustainable as a Nash equilibrium
if multiple insurance companies employ a similar test in the market.1
The remainder of this article is organized as follows. In Section 2, a brief overview of
the literature on screening is presented. Competitive insurance markets and active-screening
are characterized in Section 3. In Section 4, we investigate equilibrium in several marketsettings where insurance companies use symmetric tests which reveal both high-risk and
low-risk individuals risk type with the same probability. To establish robustness and to
provide explicit conditions, the case of asymmetric tests which perfectly identify low-risks
1Our observation is that while insurers employ similar tests, for instance most use gender in personal
lines of coverages, the classification schemes are unique across insurers.
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is discussed in Section 5. A summary of our findings and a discussion of limitations in our
work are contained in Section 6.
2 Existing studies
Screening models in a competitive environment have been considered by many others since
the seminal work of Rothschild and Stiglitz (1976). A rich literature of competitive screening
models explores possible equilibria under various assumptions (see, for instance, Miyazaki,
1977; Riley, 1979; Spence, 1978; Wilson, 1977). Recent literature has considered refinement
of the RS model and resulting equilibria (e.g., Dubey and Geanakoplos, 2002; Martin, 2007;
Netzer and Scheuer, 2010). For instance, Netzer and Scheuer (2010) consider a model where
individuals are heterogeneous in their wealth and argue that equilibria in their model are
more consistent with empirical findings. Most studies extending the RS model have relied on
the self-selection mechanism for risk sorting. Typically efforts made by uninformed parties
to acquire knowledge regarding unobserved information are not considered.
Prior studies on information acquisition have primarily dealt with constructs where con-
sumers are unaware of their risk type ex ante and invest in learning their risk type (see,
for instance, Crocker and Snow, 1992; Doherty and Thistle, 1996; Hoy and Polborn, 2000;
Polborn, Hoy, and Sadanand, 2006). In contrast, consistent with the RS model, our analysis
assumes that individuals know their risk type, but insurance companies do not.
The assumption that test results are public information also sets this study apart from
earlier work on information acquisition by Doherty and Thistle (1996) and Hoy and Pol-
born (2000). Their work assumes that tests such as genetic tests are independent from the
purchase of an insurance contract and that insurers do not have access to the test results.
This article follows research by Browne and Kamiya (2010) which investigates the market
outcome when a noisy underwriting test is offered by the insurer. Their study focuses on
identifying equilibrium conditions when insurance companies are non-myopic and all insurers
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have access to an identical underwriting test. This study differs from theirs by focusing on
Nash competition and examining the case where each insurers test is unique.
3 Competitive Insurance Market
Homogeneous risk-neutral insurers offer one type of contract defined by coverage quantity,
q, and premium, p. Individuals live in one period with initial endowment W > 0. Terminal
wealth is uncertain with two states: W1 with a fixed loss D > 0 and W2 with no loss.
Individuals differ in their likelihood of loss, which is private information known only to the
individual. The probability that loss state W1 occurs for a high risk (H-type) is H and
that for a low risk (L-type) is L, where 0 < L < H < 1. Individuals also differ in their
observable attributes, which can be used by insurance companies to predict their risk-type.
The risk-averse individual, who has a standard risk-averse utility function (i.e., U > 0,
U < 0), applies for one contract. The L-type individuals expected utility with insurance
contract C = (p, q) is VL(C) = LU(Wp + qD)+(1L)U(Wp). The proportion of
H-type individuals and that of L-type individuals in the market are public information and
are denoted by and 1 , respectively.
The classic screening models well describe the market outcomes where there is no public
information expected to be associated with private information. However, in market trans-
actions an uninformed party tends to observe informed parties characteristics. In insurance
transactions, individual characteristics such as age, gender, and marital status may be rel-
atively easily obtained by uninformed insurance companies. It is natural in such a market
that competing insurance companies immediately attempt to process a set of observed in-
formation to predict individual risk type. If one firm successfully develops an algorithm to
predict risk type more accurately than other firms, the firm will gain a market advantage.
Consider such an algorithm as a function f : X {H-type, L-type} where X represents
a set of observable attributes. The output correctly predicts the individuals risk type with
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probability y (0, 1) but fails with probability 1 y. If a firm invents a test, it has
a choice to offer either an unconditional contract or a conditional contract that may be
purchased by individuals depending on their test outcome. Thus, a conditional contract
requires individuals to take a test first and the test result determines if they can purchase
the contract.
Consider a sequential game in which a new firm decides to enter the existing insurance
market by offering contracts. When individuals maximize their expected utility, Cournot-
Nash equilibrium identified in this game can be characterized by no contract in the equilib-
rium making a loss and no contract outside the equilibrium making a non-negative profit if
offered.
In the absence of any test, the well-known RS separating equilibrium identified as a set
of contracts (H, L) holds (see Figure 1) if and only if the population of H-type individuals
is sufficiently large relative to that of L-type individuals as pointed out by Rothschild and
Stiglitz (1976). We follow Crocker and Snow (1985) and denote the RS critical value of the
proportion of H-type individuals by RS, with which the RS equilibrium holds if and only
if the actual proportion of H-type individuals is greater than or equal to the critical value
(i.e., RS). In other words, the critical value is defined where L-type individuals are
indifferent between a pooling contract with RS and the RS separating contract L. Let M be
the pooling contract which satisfies the equality, VL(M) = VL(L), where the fair premium
rate for the pooling contract is:
M = L + RS(H L) (1)
[Insert Figure 1 Here]
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4 Insurance Market with Active Screening
To investigate the possibility that individuals demand a conditional contract which upsets
the RS separating equilibrium in the case of RS, we first consider H-type individuals
demand for a conditional contract followed by L-type individuals demand. H-type individu-
als are willing to deviate from their unconditional contract H to a conditional contract, A, if
and only if EVH(A, H) VH(H), assuming that individuals choose a conditional contract
when they are indifferent between these two contracts. The left hand side of the inequality
represents the H-type individuals expected utility attained by taking the test. If the test
misclassifies one into a L-type, contract A, defined by the L-type individuals optimal pool-
ing contract at its actuarially fair premium pA, can be purchased. Otherwise, one takes the
unconditional contract H. The light hand side of the condition is the utility corresponding
to the RS separating contract H. This condition is explicitly expressed as:
yU(W HD) + (1 y)[HU(WpA + qA D) + (1 H)U(WpA)]
U(W HD)(2)
which can be reduced to VH
(A) VH
(H). This inequality holds when L-type individuals
prefer the conditional contract A to their separating contract L given the conditional contract
is offered. That is, whenever L-type individuals deviate from the unconditional contract L to
the conditional contract A, H-type individuals also deviate from the unconditional contract
H to conditional contract A. This result confines our attention to the L-types demand.
Lemma 4.1. L-type individuals prefer the conditional contract A to their unconditional
contract L in the case of
RS
if and only if the accuracy of a test employed for theconditional contract satisfies:
(1 y)
(1 y) + y(1 ) RS (3)
Proof. The conditional contract A requires a subsidy s defined by the firms resource con-
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straint where its income is equal to the expected claim payment:
y(1 )AqA + (1 y)AqA = y(1 )LqA + (1 y)HqA (4)
where A = L+ s. The subsidy rate is proportional to coverage because the subsidy paid by
L-type individuals to misclassified individuals depends on the optimal coverage determined
by maximizing the L-types utility for contract A. Then the subsidy rate can be rewritten
as:
s =H L
1 + y(1)(1y)
. (5)
Note that the RS critical value RS
is determined where L-type individuals are indifferentbetween a pooling contract with RS and its separating contract L (i.e. VL(M) = VL(L)).
Hence the L-types demand condition, VL(A) VL(L), can be restated as VL(A) VL(M).
Since the subsidy rate ofA and that ofM are s and RS(HL), respectively, the condition
is equivalent to:
H L
1 + y(1)(1y)
RS(H L). (6)
which is rearranged, in (3), by the relationship between the fraction of H-type individuals in
the pooling contract A and the RS critical value. The minimum required accuracy of a test
decreases to 0.5 as RS.
Thus both L-type and H-type individuals no longer choose their separating contracts that
fully rely on their self-selection when a test can reduce the fraction of H-type individuals in a
pooling contract to at least the RS critical value. They instead prefer a conditional contract
that utilizes a test.
When < RS, the underlying market fully relying on self-selection is characterized by
non-existence of Nash equilibrium. Individuals prefer a pooling contract W (a Wilson pooling
contract) at the average fair premium pW, to their separating contracts. It is straightforward
to show that individuals are better off with the conditional contract, A, than the uncondi-
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tional pooling contract, W, if the test accuracy is greater than 0.5. However, this accuracy
does not necessarily guarantee that insurance companies cannot offer non-equilibrium un-
conditional contracts that attract only L-type individuals given the conditional contract A
is offered. In the next section, we identify a robust condition as a part of our equilibrium
argument in the case of < RS.
4.1 Monopoly on Conditional Contracts
We start with the case where a test f is invented by one of the competing insurance compa-
nies and only the insurance company can offer a conditional contract based on that test. In
contrast to the market discussed by Stiglitz (1977), the market analyzed here is still com-petitive in that other insurance companies can offer an unconditional contract. Therefore,
in the market where RS, H-type and L-type individuals can at least attain the RS
separating contracts, H and L, respectively.
4.1.1 Equilibrium
When individuals choose a monopolists conditional contract A, L-type individuals who are
misclassified by the test and H-type individuals who are correctly identified by the test take
unconditional contracts. Therefore, a set of contracts consisting of a conditional contract
and two unconditional RS separating contracts, (A,L,H), may hold in equilibrium when
RS. In contrast, when < RS, unconditional contract W never holds in equilibrium,
while the conditional contract A may not be upset even in that case. Equilibrium holds if
no new unconditional contract can earn a non-negative profit given the conditional contract
is offered.
Although the conditional contract A is also a pooling contract, the contract can be
sustained in equilibrium. To understand this possibility, it is important to note that an
individuals decision whether to take a conditional contract is based on its ex-ante utility
for the conditional contract A, which could result in either of two contracts: A and L for
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L-type individuals and A and H for H-type individuals when RS. This is because
L-type individuals take their unconditional separating contract L if they are misclassified
as H-types by a noisy test and H-type individuals demand their unconditional separating
contract H if a noisy test correctly identifies H-type individuals as H-types. The H-types
expected utility is strictly less than the level of utility gained solely by the contract A,
EVH(A, H) < VH(A). Analogously, L-type individuals expected utility is strictly less than
the level of utility gained solely by the contract A, EVL(A, L) < VL(A).
Thus, the individuals ex-ante utility is strictly lower than its utility gained solely from
the conditional contract A. It can be shown that the deviation of the H-type individuals
expected utility from VH(A) is positively associated with the accuracy of a test and makes it
possible that no unconditional contract that attracts only L-type individuals can be offered.
It is clear that EVH(A, H) VH(H) while EVL(A, L) VL(A) as y 1 (see Figure 1).
Lemma 4.2. A set of contracts (A,L,H) holds in equilibrium when RS if there exists
an unconditional contract L defined by VH(L) = EVH(A, H) which satisfies:
EVL(A, L) VL(L) (7)
where the contract L is offered at L-types fair premium pL. In equilibrium, a conditional
contract and unconditional contracts coexist when the test employed by the conditional
contract is relatively accurate.
Lemma 4.3. Conditional contract A holds in equilibrium when < RS if there exists an
unconditional contract L defined by VH(L) = EVH(A, W) which satisfies
EVL(A, W) VL(L) (8)
These incentive conditions may be thought problematic because unconditional pooling
contract W does not hold in equilibrium. However, the Wilson pooling contract W can be
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used to define the constraint. This is because the contract W yields lower utility for L-type
and greater utility for H-type individuals than any new potential unconditional contract
offer that attracts only L-type individuals (see contract B in Figure 2). Thus, the incentive
constraints are robust against the concern of non-existence of equilibrium for the pooling
contract W, though it is still true that the unconditional pooling contract itself does not
hold in equilibrium.
[Insert Figure 2 Here]
Proposition 4.1 (Active Screening Equilibrium). A conditional pooling contract may
exist in equilibrium regardless of the proportion of H-type individuals, if an insurer with a
unique underwriting test that is sufficiently accurate offers a conditional contract.
This argument can be established by the lemmas discussed above.
4.1.2 Optimal Contract
Given that an equilibrium exists, we next consider the rent earned by a monopolist, the
single best underwriter in the market, in the case where RS. We denote the profit-
maximizing contract as . Clearly, the optimal contract is obtained where the L-types
incentive constraint holds with the equality, EVL(, L) = VL(L). Otherwise, there exists a
contract (p, q
) such that
p
+ (1 )(p
q
) > p + (1 )(p q) (9)
The optimal contract may be seen diagrammatically in Figure 3. Due to the test, the
monopolists breakeven point is at A = L + s as discussed above. The contract that
maximizes profits must lie on the indifference curve VL() and on the line parallel to the
line pA. The reason is that the vertical distance between the indifference curve VL() and
the line pA represents the profit which is maximized where a line shifted upward from pA
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is tangent to the indifference curve. Let p = (L + s)q + be the tangent line where
represents a profit per insured. Thus, the profit-maximizing contract is identified as
the tangent point of the line p to the indifference curve VL(). Note that p implies
that q = qA. Therefore, the conditional contract offered to those classified as a L-type is
= (p, qA) where p = (L + s)qA + .
[Insert Figure 3 Here]
The profit also represents the value of a noisy test, which may be sold to other firms.
A conditional contract, however, can hold in equilibrium only when the monopolist alone
offers the conditional contract. We show later that there is no equilibrium for conditional
contracts if the same test is used by other competing insurance companies.
4.2 Competitive Market with Multiple Tests
4.2.1 Heterogeneous Tests
We investigate the impact of firm competition in developing a better test to gain a com-
petitive advantage. Our particular interest here is whether a conditional contract at the
actuarially fair premium still holds in equilibrium. Consider a simple case where firm i in-
vents a test fi : X {H-type, L-type} with probability yi where i = 1, 2. Assume that
y1 > y2 > 0. Thus, one firm has a more accurate test than the other. Firm i can offer
a conditional contract Ai for those identified as a L-type by its test at the fair premium.
Hence, firm 1 offers a conditional contract A1 = (pA1, qA1) where:
pA1 = qA1
L + H L
1 + y1(1)(1y1)
(10)
A question is whether the conditional contract A1 can be sustained when both another
conditional contract and unconditional contracts are offered. Our primary concern here is
another conditional contract, A2, potentially offered by firm 2 (see Figure 4).
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[Insert Figure 4 Here]
For the conditional contract A1 to be sustained against any new conditional contract A2, the
new contract must result in an immediate loss if it is offered. This condition is rephrased as:
if a conditional contract A2 attracts L-type individuals, it must attract H-type individuals
as well. That is, the following two inequalities must be satisfied simultaneously.
EVL(A2, L) EVL(A1, L) (11)
EVH(A2, H) EVH(A1, H) (12)
To analyze these conditions, it should be noted that by definition the test for contract A2
is less accurate than the test for existing contract A1. Therefore, if firm 2 offers a conditional
contract A1 with test accuracy y2, only H-type individuals deviate from A1 to A2. Thus, in
order to avoid attracting H-type individuals, firm 2 may offer a combination of coverage and
premium which at least offsets the H-types benefit gained by the less accurate test, but still
attract L-type individuals.
Proposition 4.2. A set of contracts (A1, L , H ) holds in equilibrium in a market where firms
offer conditional pooling contracts if there is no new contract A2 that satisfies both:
EVL(A2, L) EVL(A1, L) (13)
EVH(A2, H) < EVH(A1, H) (14)
As long as the accuracy of the second best test satisfies (13) and (14), a conditional contract
A1 cannot be upset. The accuracy of a test required for a conditional contract to survive in
equilibrium cannot be explicitly derived in this case. The explicit form under the assumption
of an asymmetric test is derived in the next section. This argument can be extended to the
case where more than two insurance companies offer tests with different degrees of accuracy.
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4.2.2 Homogeneous Tests
In contrast to the previous case, we now consider the case in which multiple firms employ
the same test. Each firm offers a conditional contract A to those who are identified as a
L-type. We find that the conditional contract does not hold as a part of equilibrium.
It can be shown that there exist possible profitable deviations against a new conditional
pooling contract A. Given that contract A is offered, insurance companies may offer a
conditional contract that attracts only L-type individuals but not misclassified H-type in-
dividuals (a contract such as A2 in Figure 4). Such a conditional contract always exists
as long as L-types individuals subsidize misclassified H-type individuals. The non-existence
of equilibrium for a conditional contract can be explained just as a pooling contract can-
not be sustained in the absence of any tests. Browne and Kamiya (2010) show that the
conditional contract holds in a Wilson equilibrium where insurance companies are assumed
to be non-myopic. Thus, equilibrium in a market where firms use the same test reverts to
the RS equilibrium argument. A set of unconditional separating contracts (H, L) holds in
equilibrium if RS. Otherwise, there is no equilibrium in the presence of conditional
contracts.
Efficiency of equilibrium is another major issue when contracts fully rely on individuals
self-selection. Since we discuss conditional contracts that could be introduced to the market
where self-selection contracts are offered, by definition both H-type and L-type individuals
are better off ex ante when conditional contracts exist in equilibrium. The allocation of
the utility gain between risk types is determined by the accuracy of a test in equilibrium.
The utility gain of H-types is maximized at the minimum required accuracy of the test that
sustains equilibrium and decreases as the accuracy increases. In contrast, L-types expected
utility monotonically increases with test accuracy.
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5 Asymmetric Test
The analysis in Section 4 leads to several conclusions: first, active-screening may be utilized
in a market where private information prevails; second, a pooling contract may hold in
equilibrium; third, a conditional contract cannot be a part of an equilibrium, if the most
accurate test can be used by more than one insurance company.
We next consider an asymmetric test, which imperfectly identifies H-types but can per-
fectly identify L-types. We show that our conclusion does not depend on the assumption
that both risk-types are correctly classified with the same accuracy. We also show that some
equilibrium conditions are explicitly provided in terms of the accuracy of a test. Specifically,
we consider a test that always correctly identifies L-type individuals as L-types while H-type
individuals are misclassified with the probability 1 y. It is clear that L-type individuals
will purchase a contract A if they take the test.
With the asymmetric test, the subsidy implicit in a conditional contract A is:
s = (H L)1 y
1 y, (15)
and the required accuracy of the test for the case when RS is defined as:
y 1 (RS/)
1 RS. (16)
As RS, the right hand side of (16) decreases to zero. When the market is characterized
by < RS, it is straightforward to show that L-type individuals always prefer a condi-
tional contract, regardless of its accuracy. Thus, the accuracy condition above is a sufficient
condition such that L-type individuals prefer a conditional contract A to any unconditional
contract offer regardless of the underlying market.
Market equilibrium when the conditional contract offered by the monopolist of a con-
ditional contract employs an asymmetric noisy test can be identified analogously to the
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symmetric test case. The argument is omitted here because any further simplification can-
not be made.
The insurers profit in this case can be expressed in terms of the L-types risk premium
for contract and L
, denoted as and L, respectively. With these risk premiums, the
profit-maximizing condition, VL() = VL(L), can be expressed as:
U(Wp A A(D qA) ) = U(W LD L). (17)
This implies that the profit per insured is
= (L ) sqA. (18)
Next, we investigate a market discussed in Section 4.2.1, in which two conditional con-
tracts are offered. A conditional contract A1 holds in equilibrium if a new conditional contract
A2 that attracts L-type individuals also attracts H-type individuals (see Figure 5). That is,
a contract A2 that satisfies VL(A2) VL(A1) must also hold EVH(A2, H) EVH(A1, H).
[Insert Figure 5 Here]
Consider a new contract offer such that H-types expected utility from a new conditional
contract A2, with which only L-type individuals are worse off, makes H-types better off by
the less accurate test y2 as before. It is straightforward to show that contract L, defined by
VL(A1) = VL(L) at L-types fair premium, maximizes L-types utility loss but still attracts
L-type individuals. Therefore, if the test employed by firm 2 is inaccurate enough to increase
H-types expected utility even for contract L
, the equilibrium condition is satisfied. Thus,
we can restate the condition as EVH(L, H) EVH(A1, H), which can also be rewritten
as:
y2 VH(L) EVH(A1, H)
VH(L) VH(H). (19)
The equilibrium condition is identified in terms of the accuracy of the second best test.
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A principle difference from the symmetric test case is observed in regards to an uncon-
ditional contract offer. When L-types are perfectly classified, no unconditional contract for
L-type individuals is offered. This also implies that, regardless of the fraction of H-type
individuals, no unconditional pooling contract is offered. Therefore, equilibrium, if it exists,
is always characterized by a set of contracts (A, H).
6 Conclusion
Competitive screening has been discussed in the absence of uninformed parties efforts to
classify informed parties. Many different forms of screening are, however, employed when
private information prevails. It is reasonable for uninformed parties to attempt to predict
informed partys private information to obtain a competitive advantage in a market. We
investigate the impact of testing on the existence of equilibrium.
This paper identifies the demand for a conditional contract and we show that individuals
deviate from their self-selection contracts to a conditional contract, which requires them to
take a test, if the test reduces the fraction of H-type individuals in a pooling contract to
lower than the RS critical value of the underlying population.
Perhaps most importantly, it is shown that a pooling conditional contract holds in equi-
librium when the most accurate test has significant competitive advantage and can attract
all individuals. In contrast, we also find that conditional contracts do not hold in equilibrium
when multiple firms utilize tests similar in terms of their accuracy.
Acknowledgements
We thank Michael Hoy and seminar participants at the 2010 Risk Theory Seminar for helpful
comments. All errors are our own.
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Economic Theory 16 (2), 167207.
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W2
pLEVL(A,L)
EVH(A,H)
pA
A
H L
VH(H)
045 degree
pH L
E
W1
Figure 1: Demand of Conditional Contract A
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W2
pLEVL(A,W)
EVH(A,W) A
A
H
WLB
0 45 degree
pH L
E
W1
Figure 2: Conditional Contract A in Equilibrium
pH
H
p
pA
EV (,H)
VL()
pL
A
q
D
EVL(,L)
Figure 3: Profit-maximizing Contract
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22/22
W2
pL
EVL(A1,L)
EVH(A1,H)
pA
A1A2
VH(A1)
H L
0 45 degree
E
W1
Figure 4: Multiple Conditional Contracts
W2
pLVL(A1)
EVH(A1,H)
pA
A1A2
VH(A1)
H L
L
0 45 degree
E
W1
Figure 5: Conditional Contract A with Asymmetric Tests